Various Orders and Drawings of Plane Graphs

1. Various Orders and Drawings of Plane Graphs Takao Nishizeki Tohoku University

2. Vertex ordering 16 13 14 12 15 11 10 4 9 8 5 6 3 7 2 1

3. Vertex ordering 16 13 14 12 st-numbering 15 11 10 Canonical ordering 4 9 8 5 6 3 4-canonical ordering 7 2 Canonical decomposition 1 4-canonical decomposition

4. Vertex ordering st-numbering Canonical ordering 4-canonical ordering Canonical decomposition 4-canonical decomposition

5. t = 16 13 14 12 15 11 10 4 9 8 5 6 3 7 has two neighbors j, k 2 s = 1 st-numbering

6. has two neighbors j, k st-numbering t = 16 13 14 12 15 11 10 4 9 8 5 6 3 7 2 s = 1

7. st-numbering t = 16 13 14 12 15 11 10 4 9 8 5 6 3 7 2 s = 1 and For any i, both vertices induce connected subgraphs.

8. Application of st-numbersing Planarity testing Visibility drawing Internet routing

9. Vertex ordering st-numbering Canonical ordering 4-canonical ordering Canonical decomposition 4-canonical decomposition

10. Canonical Ordering 16 13 14 15 10 12 11 6 5 9 4 8 7 3 1 2 Triangulated plane graph

11. Canonical Ordering 16 13 14 15 10 12 11 6 5 9 4 8 7 3 1 2 Gk: subgraph of G induced by vertices

12. Canonical Ordering 16 13 14 15 10 12 11 6 G9 5 9 4 8 7 3 1 2 Gk: subgraph of G induced by vertices

13. For any Canonical Ordering 16 13 14 15 10 12 11 6 5 (co1) Gk is biconnected and internally triangulated 9 4 8 7 3 1 2 (co2) vertices 1 and 2 are on the outer face of Gk (co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

14. For any Canonical Ordering 16 13 14 15 10 12 11 6 5 (co1) Gk is biconnected and internally triangulated 9 4 8 7 3 1 2 G3 (co2) vertices 1 and 2 are on the outer face of Gk (co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

15. For any Canonical Ordering 16 13 14 15 10 12 11 6 5 (co1) Gk is biconnected and internally triangulated 9 4 8 7 3 1 2 G4 (co2) vertices 1 and 2 are on the outer face of Gk (co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

16. For any Canonical Ordering 16 13 14 15 10 12 11 6 5 (co1) Gk is biconnected and internally triangulated 9 4 8 7 3 1 2 G10 (co2) vertices 1 and 2 are on the outer face of Gk (co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

17. For any Canonical Ordering 16 13 14 15 10 12 11 6 5 (co1) Gk is biconnected and internally triangulated 9 4 8 7 3 1 2 G10 (co2) vertices 1 and 2 are on the outer face of Gk (co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

18. For any Canonical Ordering 16 13 14 15 10 12 11 6 5 (co1) Gk is biconnected and internally triangulated 9 4 8 7 3 1 2 G10 (co2) vertices 1 and 2 are on the outer face of Gk (co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

19. Straight Line Grid Drawing Straight line grid drawing. Plane graph de Fraysseix et al.’90

20. Initial drawing of G3

21. Install k + 1 Gk

22. Shift and install k + 1 Shift method

23. Schnyder ’90 H n-2 n-2 W Upper bound

24. What is the minimum size of a grid required for a straight line drawing?

25. Lower Bound

26. A restricted class of plane graphs may have more compact grid drawing. Triangulated plane graph 3-connected graph

27. 4-connected ? disconnected not 4-connected

28. How much area is required for 4-connected plane graphs?

29. Straight line grid drawing Miura et al. ’01 Input: 4-connected plane graph G Output: a straight line grid drawing Grid Size : H <n/2 = Area: W < n/2 =

30. Area < 1/4 = Miura et al. ’01 Schnyder ’90 4-connected plane graph G plane graph G n-2 H <n/2 = W < n/2 n-2 = . Area=n Area<n /4 . 2 2 =

31. Vertex ordering st-numbering Canonical ordering 4-canonical ordering Canonical decomposition 4-canonical decomposition

32. Triangulate all inner faces Step1: find a 4-canonical ordering G” n=18 n-1=17 n=18 n-1=17 16 16 Step2: Divide G into two halvesG’ and G” 15 15 12 12 14 14 10 10 13 13 11 11 9 n/2=9 8 8 5 5 6 6 7 7 4 4 Step3 and 4 : Draw G’ and G” in isosceles right-angled triangles 3 3 1 2 1 2 G’ G” W/2 G 1 |slope|>1 |slope|>1 |slope|>1 Step5: Combine the drawings of G’ and G” n/2 W/2 |slope|<1 = G’ W < n/2 -1 n/2 -1 = Main idea G

33. 4-canonical ordering [KH97](4-connected graph) (1) Edges (1,2) and (n,n-1) are on the outer face (2) For each vertex k, 3 < k < n-2, at least two neighbors have lower number and at least two neighbors have higher neighbor. n=18 n-1=17 16 15 12 14 10 13 11 9 8 5 6 7 4 3 2 1

34. 4-canonical ordering [KH97](4-connected graph) n=18 n-1=17 16 15 12 14 Generalization of an st-numbering 10 13 11 9 8 5 6 7 4 3 2 1 Both vertices and induce 2-connected subgraphs.

35. Shift and install k + 1 Shift method Only one shift

36. Shift and install k + 1 Shift method

37. Graph is not triangulated Is there any ordering?

38. Vertex ordering st-numbering Canonical ordering 4-canonical ordering Canonical decomposition 4-canonical decomposition

39. u u u n 1 2 Canonical Decomposition

40. V V V V V V V V 4 8 7 6 5 3 2 1 u u u n 1 2 Canonical Decomposition

41. V V V V V V V V 3 5 7 4 2 8 6 1 u u u n 1 2 Canonical Decomposition (cd1) V1 is the set of all vertices on the inner face containing edge (u1,u2)． (cd2) for each index k, 1 ≤ k ≤ h, Gkis internally 3-connected. (cd3) for each k, 2≦k≦h-1, vertices in Vk are on the outer vertices and the following (a) and (b) holds．

42. V V V V V V V V 3 5 7 4 2 8 6 1 u u u n 1 2 Canonical Decomposition (cd1) V1 is the set of all vertices on the inner face containing edge (u1,u2)． (cd2) for each index k, 1 ≤ k ≤ h, Gkis internally 3-connected. (cd3) for each k, 2≦k≦h-1, vertices in Vk are on the outer vertices and the following (a) and (b) holds．

43. separation pair u v Internally 3-connected G is bi-connected For any separation pair {u,v} of G uand vare outer vertices each connected component of G-{u,v} contains an outer vertex.

44. Internally 3-connected G is bi-connected For any separation pair {u,v} of G uand vare outer vertices each connected component of G-{u,v} contains an outer vertex.

45. v 2 u separation pair v Internally 3-connected G is bi-connected For any separation pair {u,v} of G uand vare outer vertices each connected component of G-{u,v} contains an outer vertex.

46. Internally 3-connected G is bi-connected For any separation pair {u,v} of G uand vare outer vertices each connected component of G-{u,v} contains an outer vertex.

47. Internally 3-connected G is bi-connected For any separation pair {u,v} of G uand vare outer vertices each connected component of G-{u,v} contains an outer vertex.

48. Internally 3-connected G is bi-connected For any separation pair {u,v} of G uand vare outer vertices each connected component of G-{u,v} contains an outer vertex.

49. Internally 3-connected G is bi-connected For any separation pair {u,v} of G uand vare outer vertices each connected component of G-{u,v} contains an outer vertex.

50. Internally 3-connected G is bi-connected For any separation pair {u,v} of G uand vare outer vertices each connected component of G-{u,v} contains an outer vertex.